472 I>B- A - N. WHITEHEAD ON 



of being a class possessing at least one member; and (x) . $lx means $\x is true for 

 every value of x; and fax) . <j>\x means there exists a value of x for which <f>lx is 

 true. Note that (x) and fax), written before any proposition involving x, give the 

 above meanings, even if the proposition is not in the symbolic form <f>\x. If the 

 proposition involve both x and y, then (x, y) prefixed means that the proposition is 

 true for all values of x and y ; and similarly for (x, y, z), and so on. Also (gee, y) 

 prefixed means that there exist values of x and y, such that the proposition is true ; 

 and similarly for fax, y, z), and so on. Furthermore <j>\ x -D X \jj\x stands for 

 (x) . {(f)lx 3 \)flx}, and similarly for two and three variables. 



On the Use of Dots, viz., ., :, .'., : : 



p . q or p : q or p . '. q or p : : q all mean p and q are both true propositions. As 

 an example, x, y e u, which has been defined above, is really the proposition 

 xeu.yeii', and x, y, z e u is the proposition xeu . y eu . zeu. 



Dots as Brackets. The different symbolic forms for the joint assertion of pro- 

 positions arise from the fact that dots are also used as bracket forms for propositions 

 according to the following rules : 



(i) The larger aggregation of dots represents the exterior bracket, (ii) The dots 

 at the end of a complete sequence of symbols are omitted, (iii) The dots immediately 

 preceding or succeeding the implication sign, viz., D, are exterior brackets to any 

 equal number of dots occurring in other capacities (e.g., as above in the joint assertion 

 of propositions), (iv) The dots which also serve to indicate the joint assertion of 

 propositions are interior brackets to any equal number of dots occurring in other 

 capacities, (v) The dots after (x) and fax) are increased in number according to the 

 necessity for their use as brackets. 



In reading a symbolic proposition it is best to begin by searching for that 

 implication sign, viz., 3, which is preceded or succeeded by the greatest number of 

 dots. This splits up the proposition into hypothesis and consequent ; and so on with 

 these subsidiary propositions, if necessary. 



On V, *, ^e, ?, ^(x).4>lx, -^fax).<t>\x 



Again >V? means one or other or each of p and q is a true proposition; and -^-p 

 means p is not true. Thus ~- <j> ! x means x has not the property <j> ; also x -~ e u 

 stands for (x eu); and x ^ y stands for ^-(x = y); and -^(x) . <f>lx stands for 

 ^ {(x) . <j)lx} ; and ^fax) . <f)\x stands for ^ {fax) . 



Onx(<f>!x), (ix)(<l>\x), i', 7', u, n, u', n' 



Non-Propositional Functions. x(<j>\x) denotes the class of terms which have the 

 property (ft, and (ix) (<f>\x) denotes the single entity (if there is such) which, when 



